How to get close to the median shape

Abstract

In this paper, we study the problem of L1-fitting a shape to a set of n points in R d (where d is a fixed constant), where the target is to minimize the sum of distances of the points to the shape, or alternatively the sum of squared distances. We present a general technique for computing a (1+)-approximation for such a problem, with running time O(n+poly(logn,1/)), where poly(logn,1/) is a polynomial of constant degree of logn and 1/ (the power of the polynomial is a function of d). This is a linear time algorithm for a fixed >0, and is the first subquadratic algorithm for this problem. Applications of the algorithm include best fitting either a circle, a sphere or a cylinder to a set of points when minimizing the sum of distances (or squared distances) to the respective shape.

abstract = "In this paper, we study the problem of L1-fitting a shape to a set of n points in R d (where d is a fixed constant), where the target is to minimize the sum of distances of the points to the shape, or alternatively the sum of squared distances. We present a general technique for computing a (1+)-approximation for such a problem, with running time O(n+poly(logn,1/)), where poly(logn,1/) is a polynomial of constant degree of logn and 1/ (the power of the polynomial is a function of d). This is a linear time algorithm for a fixed >0, and is the first subquadratic algorithm for this problem. Applications of the algorithm include best fitting either a circle, a sphere or a cylinder to a set of points when minimizing the sum of distances (or squared distances) to the respective shape.",

keywords = "Approximation algorithms, Shape fitting",

author = "Sariel Har-Peled",

year = "2007",

month = jan

day = "1",

doi = "10.1016/j.comgeo.2006.02.003",

language = "English (US)",

volume = "36",

pages = "39--51",

journal = "Computational Geometry: Theory and Applications",

issn = "0925-7721",

publisher = "Elsevier",

number = "1",

}

TY - JOUR

T1 - How to get close to the median shape

AU - Har-Peled, Sariel

PY - 2007/1/1

Y1 - 2007/1/1

N2 - In this paper, we study the problem of L1-fitting a shape to a set of n points in R d (where d is a fixed constant), where the target is to minimize the sum of distances of the points to the shape, or alternatively the sum of squared distances. We present a general technique for computing a (1+)-approximation for such a problem, with running time O(n+poly(logn,1/)), where poly(logn,1/) is a polynomial of constant degree of logn and 1/ (the power of the polynomial is a function of d). This is a linear time algorithm for a fixed >0, and is the first subquadratic algorithm for this problem. Applications of the algorithm include best fitting either a circle, a sphere or a cylinder to a set of points when minimizing the sum of distances (or squared distances) to the respective shape.

AB - In this paper, we study the problem of L1-fitting a shape to a set of n points in R d (where d is a fixed constant), where the target is to minimize the sum of distances of the points to the shape, or alternatively the sum of squared distances. We present a general technique for computing a (1+)-approximation for such a problem, with running time O(n+poly(logn,1/)), where poly(logn,1/) is a polynomial of constant degree of logn and 1/ (the power of the polynomial is a function of d). This is a linear time algorithm for a fixed >0, and is the first subquadratic algorithm for this problem. Applications of the algorithm include best fitting either a circle, a sphere or a cylinder to a set of points when minimizing the sum of distances (or squared distances) to the respective shape.